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Nicod & Hempel in Order to Fully Understand Goodman's “New Riddle,”

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Nicod & Hempel in Order to Fully Understand Goodman's “New Riddle,” 2 BRANDEN FITELSON Confirmation consists in positive probabilistic relevance, and disconfirma- • tion consists in negative probabilistic relevance (where the salient probabil- ities are inductive in the Keynesian [21] sense). GOODMAN’S “NEW RIDDLE” Universal generalizations are confirmed by their positive instances and dis- • BRANDEN FITELSON confirmed by their negative instances. Hempel [17] offers a precise, logical reconstruction of Nicod’s naïve instantial ac- Abstract. First, a brief historical trace of the developments in confirmation the- count. There are several peculiar features of Hempel’s reconstruction of Nicod. I ory leading up to Goodman’s infamous “grue” paradox is presented. Then, Good- will focus presently on two such features. First, Hempel’s reconstruction is com- man’s argument is analyzed from both Hempelian and Bayesian perspectives. pletely non-probabilistic (we’ll return to that later). Second, Hempel’s reconstruc- A guiding analogy is drawn between certain arguments against classical deduc- tive logic, and Goodman’s “grue” argument against classical inductive logic. The tion takes the relata of Nicod’s confirmation relation to be objects and universal upshot of this analogy is that the “New Riddle” is not as vexing as many com- statements, as opposed to singular statements and universal statements. In mod- mentators have claimed (especially, from a Bayesian inductive-logical point of ern (first-order) parlance, Hempel’s reconstruction of Nicod can be expressed as: view). Specifically, the analogy reveals an intimate connection between Good- man’s problem, and the “problem of old evidence”. Several other novel aspects (NC0) For all objects x (with names x), and for all predicate expressions φ and ψ: of Goodman’s argument are also discussed (mainly, from a Bayesian perspective). x confirms [( y)(φy ψy)\ iff [φx & ψx\ is true, and ∀ ⊃ 1 x disconfirms [( y)(φy ψy)\ iff [φx & ψx\ is true. ∀ ⊃ ∼ 1. Prehistory: Nicod & Hempel As Hempel explains, (NC0) has absurd consequences. For one thing, (NC0) leads to a theory of confirmation that violates the hypothetical equivalence condition: In order to fully understand Goodman’s “New Riddle,” it is useful to place it (EQC ) If confirms , then confirms anything logically equivalent to . in its proper historical context. As we will see shortly, one of Goodman’s central H x H x H aims (with his “New Riddle”) was to uncover an interesting problem that plagues To see this, note that — according to (NC0) — both of the following obtain: Hempel’s [17] theory of confirmation. And, Hempel’s theory was inspired by some a confirms “( y)(Fy Gy),” provided a is such that Fa & Ga. of Nicod’s [25] earlier, inchoate remarks about instantial confirmation, such as: • ∀ ⊃ Nothing can confirm “( y)[(Fy & Gy) (Fy & Fy)],” since no object Consider the formula or the law: A entails B. How can a particular propo- • ∀ ∼ ⊃ ∼ a can be such that Fa & Fa. sition, [i.e.] a fact, affect its probability? If this fact consists of the pres- ∼ ence of B in a case of A, it is favourable to the law . on the contrary, if it But, “( y)(Fy Gy)” and “( y)[(Fy & Gy) (Fy & Fy)]” are logically consists of the absence of B in a case of A, it is unfavourable to this law. ∀ ⊃ ∀ ∼ ⊃ ∼ equivalent. Thus, (NC0) implies that a confirms the hypothesis that all Fs are Gs By “is (un)favorable to,” Nicod meant “raises (lowers) the inductive probability of”. only if this hypothesis is expressed in a particular way. I agree with Hempel that this In other words, Nicod is describing a probabilistic relevance conception of confir- violation of (EQCH ) is a compelling reason to reject (NC0) as an account of instantial mation, according to which it is postulated (roughly) that positive instances are confirmation.2 But, I think Hempel’s reconstruction of Nicod is uncharitable on probability-raisers of universal generalizations. Here, Nicod has in mind Keynesian this score (more on that later). In any event, my main focus here will be on the [21] inductive probability (more on that later). While Nicod is not very clear on theory Hempel ultimately ends-up with. For present purposes, there is no need the logical details of his probability-raising account of instantial confirmation (stay to examine Hempel’s [17] ultimate theory of confirmation in its full logical glory. tuned for both Hempelian and Bayesian precisifications/reconstructions of Nicod’s For our subsequent discussion of Goodman’s “New Riddle,” we’ll mainly need just remarks), three aspects of Nicod’s conception of confirmation are apparent: the following two properties of Hempel’s confirmation relation, which is a relation Instantial confirmation is a relation between singular and general proposi- between statements and statements, as opposed to objects and statements.3 • tions/statements (or, if you will, between “facts” and “laws”). 1 Date: 01/02/08. Penultimate draft (final version to appear in the Journal of Philosophical Logic). Traditionally, such conditions are called “Nicod’s Criteria”, which explains the abbreviations. I would like to thank audiences at the Mathematical Methods in Philosophy Workshop in Banff, the Historically, however, it would be more accurate to call them “Keynes’s Criteria” (KC), since Keynes Reasoning about Probabilities & Probabilistic Reasoning conference in Amsterdam, the Why Formal [21] was really the first to explicitly endorse the three conditions (above) that Nicod later championed. 2 Epistemology? conference in Norman, the philosophy departments of New York University and the Not everyone in the literature accepts (EQCH ), especially if (EQCH ) is understood in terms of University of Southern California, and the Group in Logic and the Methodology of Science in Berkeley classical logical equivalence. See [28] for discussion. I will simply assume (EQCH ) throughout my for useful discussions concerning the central arguments of this paper. On an individual level, I am discussion. I will not defend this assumption (or its evidential counterpart, which will be discussed indebted to Johan van Benthem, Darren Bradley, Fabrizio Cariani, David Chalmers, Kenny Easwaran, below), which I take to be common ground in the present historical dialectic. 3 Hartry Field, Alan Hájek, Jim Hawthorne, Chris Hitchcock, Paul Horwich, Franz Huber, Jim Joyce, (NC) is clearly a more charitable reconstruction of Nicod’s (3) than (NC0) is. So, perhaps it’s most Matt Kotzen, Jon Kvanvig, Steve Leeds, Hannes Leitgeb, John MacFarlane, Jim Pryor, Susanna Rinard, charitable to read Hempel as taking (NC) as his reconstruction of Nicod’s remarks on instantial confir- Jake Ross, Sherri Roush, Gerhard Schurz, Elliott Sober, Michael Strevens, Barry Stroud, Scott Stur- mation. In any case, Hempel’s reconstruction still ignores the probabilistic aspect of Nicod’s account. geon, Brian Talbot, Mike Titelbaum, Tim Williamson, Gideon Yaffe, and an anonymous referee for the This will be important later, when we discuss probabilistic approaches to Goodman’s “New Riddle”. Journal of Philosophical Logic for comments and suggestions about various versions of this paper. At that time, we will see an even more charitable (Bayesian) reconstruction of Nicod’s remarks. GOODMAN’S “NEW RIDDLE” 3 4 BRANDEN FITELSON (NC) For all constants x and for all (independent) predicate expressions φ, ψ: (H2) All emeralds are grue. More formally, H2 is: ( x)(Ex Gx). ∀ ⊃ [φx & ψx\ confirms [( y)(φy ψy)\ and Even more precisely, H2 is: ( x)[Ex (Ox Gx)]. ∀ ⊃ ∀ ⊃ ≡ [φx & ψx\ disconfirms [( y)(φy ψy)\. ∼ ∀ ⊃ And, we can state three salient singular (evidential) statements, about an object a: (M) For all constants x, for all (consistent) predicate expressions , , and for φ ψ ( 1) Ea & Ga.[a is an emerald and a is green] all statements H: If [φx\ confirms H, then [φx & ψx\ confirms H. E ( 2) Ea & (Oa Ga).[a is an emerald and a is grue] As we will see, it is far from obvious whether (NC) and (M) are (intuitively) correct E ≡ ( ) Ea & Oa & Ga.[a is an emerald and a is both green and grue] confirmation-theoretic principles (especially, from a probabilistic point of view!). E Severally, (NC) and (M) have some rather undesirable consequences, and jointly Now, we are in a position to look more carefully at the various claims Goodman they face the full brunt of Goodman’s “New Riddle,” to which I now turn. makes in the above passage. At the beginning of the passage, Goodman considers an object a which is examined before t, an emerald, and green. He correctly points 2. History: Hempel & Goodman out that all three of 1, 2, and are true of such an object a. And, he is also E E E Generally, Goodman [14] speaks rather highly of Hempel’s theory of confirma- correct when he points out that — according to Hempel’s theory of confirmation — 1 confirms H1 and 2 confirms H2. This just follows from (NC). Then, he seems tion. However, Goodman [14, ch. 3] thinks his “New Riddle of Induction” shows E E that Hempel’s theory is in need of rather serious revision/augmentation. Here is to suggest that it therefore follows that “the observation of a at t” confirms both an extended quote from Goodman, which describes his “New Riddle” in detail: H1 and H2 “equally.” It is important to note that this claim of Goodman’s can only be correct if “the observation of at ” is a statement (since Hempel’s rela- Now let me introduce another predicate less familiar than “green”. It is a t tion is a relation between statements and statements, not between “observations” the predicate “grue” and it applies to all things examined before t just in case they are green but to other things just in case they are blue. Then at and statements) which bears the Hempelian confirmation relation to both universal time t we have, for each evidence statement asserting that a given emerald statements H1 and H2.
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